Applications of Nijenhuis Geometry V: geodesically equivalent metrics and finite-dimensional reductions of certain integrable quasilinear systems

TL;DR

This paper characterizes metrics compatible with a Nijenhuis operator and demonstrates how these metrics lead to finite-dimensional reductions of certain integrable hydrodynamic PDEs, linking geometry with integrability.

Contribution

It provides a complete description of geodesically compatible metrics with a Nijenhuis operator and connects this to finite-dimensional reductions of integrable PDE systems.

Findings

All metrics compatible with a given Nijenhuis operator are described.

A generic curve can be a geodesic for some compatible metric.

Finite-dimensional reductions are obtained via geodesic compatibility and Poisson actions.

Abstract

We describe all metrics geodesically compatible with a gl-regular Nijenhuis operator $L$. The set of such metrics is large enough so that a generic local curve $\gamma$ is a geodesic for a suitable metric $g$ from this set. Next, we show that a certain evolutionary PDE system of hydrodynamic type constructed from $L$ preserves the property of $\gamma$ to be a $g$-geodesic. This implies that every metric $g$ geodesically compatible with $L$ gives us a finite dimensional reduction of this PDE system. We show that its restriction onto the set of $g$-geodesics is naturally equivalent to the Poisson action of $\mathbb{R}^n$ on the cotangent bundle generated by the integrals coming from geodesic compatibility.